Irreducible representations of the symmetric group Sym$(n)$, and degree-$n$ algebraic representations of SL$_d(\mathbb C)$ for $d\ge n$, can both be classified by Young diagrams with $n$ boxes.

Consider now a degree-$n$ irreducible representation of SL$_d(\mathbb C)$ with $d\ge n$, and restrict it to the subgroup of permutation matrices Sym$(d)$. What is its decomposition in irreducibles? Is there a nice way of listing those Young diagrams (with $d$ boxes) which occur, constructed out of the Young diagram with $n$ boxes of the original representation?